Vol. 121, No. 8, pp.663-748.

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## ARTICLES

### The Seventy-Fourth William Lowell Putnam Mathematical Competition

Leonard F. Klosinski, Gerald L. Alexanderson, and Mark Krusemeyer

The results of the Seventy-Fourth William Lowell Putnam Mathematical Competition.

### Carries, Group Theory, and Additive Combinatorics

Persi Diaconis, Xuancheng Shao, and Kannan Soundararajan

Given a group and a normal subgroup, we study the problem of choosing coset representatives with few “carries.” The problem is closely linked to the emerging field of additive combinatorics. We explore this link and give a gentle introduction to some results and techniques of additive combinatorics.

### On the Maximum Arc Length of Monotonic Functions

Marc Deléglise and Andrew Markoe

We revisit a problem solved in 1963 by Zaanen and Luxemburg in this *Monthly*: What is the largest possible length of the graph of a monotonic function on an interval? Is there such a function that attains this length? This is an interesting and intriguing problem with a somewhat surprising answer that should be of interest to a broad spectrum of mathematicians, starting with upper-level undergraduates. Our proof is very elementary, as opposed to the proof of Zaanen and Luxemburg. We are also able to give a pleasing geometric interpretation of our proof that is not possible with the proof of Zaanen and Luxemburg.

### Deriving New Sinc Results from Old

David Borwein and Jonathan Borwein

From previously established results in [**2**], we develop a simple proof of Keith Ball’s expression in [**1**] for the volume of the intersection of an (*n* – 1)-dimensional hyperplane with an *n*-dimensional cube, as well as a simple proof of the formula given by Frank and Riede in [**5**] for that volume.

### NOTES

### Foldable Triangulations of Lattice Polygons

Michael Joswig and Günter M. Ziegler

We give a simple formula for the signature of a foldable triangulation of a lattice polygon in terms of its boundary. This yields lower bounds on the number of real roots of certain systems of polynomial equations known as “Wronski systems.”

### Pólya’s Random Walk Theorem

Jonathan Novak

This note presents a proof of Pólya’s random walk theorem using classical methods from special function theory and asymptotic analysis.

### Segmentally Alternating Series

Michael Schramm, John Troutman, and Daniel Waterman

We consider real series composed of segments having terms of mixed sign and obtain results generalizing the alternating series theorem. We apply this to prove a result on integration of a series of functions which is a numerical series of this type almost everywhere.

### Connections Between Cubic Splines and Quadrature Rules

Mark H. Holmes

The formula for the cubic spline is integrated exactly, and this is used to derive a composite integration rule. The approach is also reversed, with the result that end conditions for the spline are derived that produce higher-order quadrature formulas.

### Hausdorffization and Such

M. Scott Osborne

If $$G$$ is a group, with commutator subgroup $$G\mathbb{Z}$$, then the quotient group $$G/g\mathbb{Z}$$ serves as an “Abelianization” of $$G$$. Any homomorphism from $$G$$ to an Abelian group factors through $$G/G\mathbb{Z}$$. Similarly, any topological space has a “Hausdorffization,” as well as several more such constructs (which is the subject of this paper).

### Binomial Coefficients Involving Infinite Powers of Primes

Donald M. Davis

If $$p$$ is a prime (implicit in notation) and $$n$$ a positive integer, let $$v(n)$$ denote the exponent of $$p$$ in $$n$$, and $$U(n)=n/p^{v(n)}$$, the unit part of $$n$$. If $$\alpha$$ is a positive integer not divisible by $$p$$, we show that the $$p$$-adic limit of $$(-1)^{p\alpha e}U((\alpha p^{e})!)$$ as $$e\rightarrow\infty$$ is a well-defined $$p$$-adic integer, which we call $$z_{\alpha}$$. Note that if $$p=2$$ or $$\alpha$$ is even, then this can be thought of as $$U((\alpha p^{\infty})!)$$. In terms of these, we give a formula for the $$p$$-adic limit of $$\left(\begin{array}{c}ap^{e}+c\\bp^{e}+d\end{array}\right)$$ as $$e\rightarrow\infty$$, which we call $$\left(\begin{array}{c}ap^{\infty}+c\\bp^{\infty}+d\end{array}\right)$$. Here $$a\geq b$$ are positive integers, and $$c$$ and $$d$$ are integers.

## PROBLEMS AND SOLUTIONS

Problems 11796–11802

Solutions 11658, 11662, 11663, 11668, 11669, 11670

## REVIEWS

*The Cult of Pythagorus: Math and Myths*. By Alberto A. Martínez.

Reviewed by Brian M. Loft

## Mathbits

### Law of Cosines—A Proof Without Words

John Molokach